Abstract
Let A be a commutative semigroup which has either a greatest regular image or a greatest group image. Then for any commutative semigroup B, A⊗B has a greatest image of the same type and it is describable by standard constructions based on A and B. If a commutative semigroup A has a greatest group-with-zero image then A⊗B has such an image if and only if B is archi-medean, in which case this image is again describable by standard constructions based on A and B. A handy elementary tool is the fact that the Grothendieck group of a commutative semigroup A may be regarded as the direct limit of the directed system of groups provided by Z⊗A where Z is the additive group of integers.
